Citation

Soyster AL, Wilson GR. Highw. Res. Rec. 1973; 456: 28-39.

Copyright

(Copyright © 1973, National Research Council (U.S.A.), Highway Research Board)

DOI

unavailable

PMID

unavailable

Abstract

In the fundamental relationship between flow and concentration, flow (in vph) increases with concentration (in vehicles/mile) until a critical point is reached. After this critical point, flow decreases to zero as concentration increases to saturation. This is a deterministic model relating flow rate q to concentration k. In this paper this deterministic model is extended by allowing a probabilistic distribution of concentrations for a given mean value of flow. The specific application is to traffic proceeding up a two-lane hill. In this stochastic model, platoons arrive at the bottom of the hill in a poisson fashion with parameter lambda and at the top of the hill in a poisson fashion with parameter mu. Because the size of platoon at the top of the hill is generally considerably larger than that at the bottom, lambda is greater than mu. The distributions of platoon sizes at both the bottom and the top of the hill are additional parameters in the formulation. Vehicles on a hill represent a birth and death process where arrival of vehicles at the bottom corresponds to births and arrival of vehicles at the top of the hill corresponds to deaths. Because the lower bound on the number of vehicles is zero and the upper bound is determined from the length of the hill and the length of vehicles, there are a finite number of possible states. These states are incorporated into a finite markov chain with a transition matrix determined by lambda, mu, and the distribution of platoon sizes at the bottom and top of the hill. The transition matrix generates the probability of various concentrations on the hill as a function of the input parameters and time t. Hence, instead of two concentrations corresponding to a mean flow rate, we generate a probability distribution that varies with time for a whole range of concentrations. The markov process also generates certain dynamic properties of the system such as relative stability. These and other stochastic properties of the markov process are included to provide an extension of the classical flow-concentration deterministic model.

Language: en

Keywords

MATHEMATICAL MODELS; TRAFFIC SURVEYS